P.Mean: A sportswriter tackles the Monty Hall problem (created 2009-03-31).

Joe Posnanski, a famous sports writer who loves statistics, wrote a couple of entries in his blog about the famous Monty Hall problem. I find the problem trite and annoying, but that probably says something more about me than about the problem. It is a very popular problem highly cited on the Internet and in many print publications. The Wikipedia quotes it from Parade magazine in 1990.

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? en.wikipedia.org/wiki/Monty_Hall_problem

For those of you unaware of the reference, Monty Hall was the host of a famous television show, Let's Make a Deal.

The answer, which is quite counter-intuitive, is that you are better off switching doors. Your probability of winning is 1/3 if you stay with your initial choice and 2/3 if you switch. This is all fine and good, and there are some nice explanations at the SCI.MATH FAQ and the REC.PUZZLES FAQ.

It is worth noting, however, that this problem makes some implicit assumptions. In particular, it relies on the belief that Monty Hall always reveals the unpicked door that has a goat behind it.

Suppose we make a different assumption, one I call the "Malevolent Monty" assumption. Suppose that Monty Hall, who knows where the car is, only reveals the goat and entices you with a switch when you have already selected the car. Then your probability of winning is 0%. That's not quite as nice as 2/3, now is it?

That's why I find these problems annoying. They always make an implicit assumption that you are supposed to figure out. When you don't figure out this implicit assumption, you're told smugly that this was an obvious part of the problem.

Creative Commons License This work is licensed under a Creative Commons Attribution 3.0 United States License. This page was written by Steve Simon and was last modified on 2010-04-12. Need more information? I have a page with general help resources. You can also browse for pages similar to this one at Category: Probability concepts.